A 60 kg skier leaves the end of a ski-jump ramp with a velocity of 26 m/s directed 23° above the horizontal. Suppose that as a result of air drag the skier returns to the ground with a speed of 19 m/s, landing 8.3 m vertically below the end of the ramp. From the launch to the return to the ground, by how much is the mechanical energy of the skier-Earth system reduced because of air drag?

I got -14330.4J, but this answer is wrong and I can't figure out why

I did KE= 09450
PE=-4880.4
ME=-14330.4

thanks

Well I get:

initial KE = (1/2)(60)(26)^2 = 20,280 J
gain due to fall = m g h = 60(9.81)(8.3) = 4485 J
so total energy at ground if no friction = 25,165 J

actual Ke at ground = (1/2)(60)(19)^2 = 10,830 J

difference = 14,335 J which is the loss due to air drag

To determine the change in mechanical energy of the skier-Earth system due to air drag, you need to calculate the initial and final mechanical energies and then find the difference. Here's how you can go about it:

1. Calculate the initial kinetic energy (KEi):
KEi = 0.5 * mass * velocity^2

In this case, the mass of the skier is 60 kg and the initial velocity is 26 m/s. So, plug these values into the formula:
KEi = 0.5 * 60 kg * (26 m/s)^2

2. Calculate the potential energy at the launch point (PElaunch):
Since the problem doesn't provide the height of the ramp, we can assume it to be zero. Therefore, the potential energy at the launch point can be set to zero as well.

PElaunch = 0

3. Calculate the potential energy at the return point (PEreturn):
The problem states that the skier lands 8.3 m vertically below the end of the ramp. So, to find the potential energy at the return point, we need to consider the change in height from the launch to the return:
Δh = -8.3 m (negative because the skier descends)

The formula for potential energy is: PE = mass * gravity * height.
The acceleration due to gravity is approximately 9.8 m/s^2.

PEreturn = mass * gravity * Δh
= 60 kg * 9.8 m/s^2 * (-8.3 m)

4. Calculate the final kinetic energy (KEf):
The problem states that the skier returns to the ground with a speed of 19 m/s. So, we can calculate the final kinetic energy using the same formula as before:
KEf = 0.5 * mass * velocity^2
= 0.5 * 60 kg * (19 m/s)^2

5. Calculate the final mechanical energy (MEf):
Since the potential energy at the return point is negative, we need to subtract it from the final kinetic energy:
MEf = KEf + PEreturn

6. Calculate the change in mechanical energy:
ΔME = MEf - MEi
= MEf - (KEi + PElaunch)

Now, let's plug in the values and calculate:

KEi = 0.5 * 60 kg * (26 m/s)^2
= 0.5 * 60 kg * 676 m^2/s^2
= 10,140 J

PElaunch = 0

PEreturn = 60 kg * 9.8 m/s^2 * (-8.3 m)
= -4,531.2 J

KEf = 0.5 * 60 kg * (19 m/s)^2
= 0.5 * 60 kg * 361 m^2/s^2
= 6,840 J

MEf = KEf + PEreturn
= 6,840 J + (-4,531.2 J)
= 2,308.8 J

ΔME = MEf - MEi
= 2,308.8 J - (10,140 J + 0)
= -7,831.2 J

Therefore, the change in mechanical energy of the skier-Earth system due to air drag is approximately -7,831.2 J.